Transcript The fermion condensations and the ηmeson in θ vacuum

Towards θvacuum simulation in
lattice QCD
Hidenori Fukaya
YITP, Kyoto Univ.
Collaboration with
S.Hashimoto (KEK), T.Hirohashi (Kyoto Univ.),
K.Ogawa(Sokendai), T.Onogi(YITP)
Contents
1.
2.
3.
4.
5.
Introduction
θ vacuum simulation of 2D QED
How to fix topology in 4D QCD
Tempering for QCD θ vacuum
Conclusion
1. Introduction
Chiral symmetry and topology are very
important and related each other...
Chiral symmetry breaking
 Anomaly
 Construction of chiral gauge theory
 4D N=1 SUSY (Chiral super fields)

Our aim is to keep chiral symmetry and
topological properties in 4D lattice QCD.
⇒ θ vacuum simulation
1. Introduction

Ginsparg-Wilson relation
Phys.Rev.D25,2649(‘82)
⇒ exact chiral symmetry at classical level.

Luscher’s “admissibility” condition
Phys.Lett.B428,342(‘98),Nucl.Phys.B549,295(‘99)
⇒ locality of Dirac operator
⇒ Stability of the index (topological charge)
1. Introduction
Our goal is to calculate path integrals with θterm;
⇒ We need
and
.
..
Without
Luscher’s
bound
“admissibility”
condition
realized
our
If
gauge
fields
are
“admissible”
ε＜
),
2.
θvacuum
simulation
of (),2D
QED
InThe
numerical
study
( using is
HMC
the2by
a gauge action
proposed
by Luscher;
topological
charge
is
actually
conserved
!!!
 The 2-flavor massive Schwinger
model
⇒ topological charge can jump； Q → Q±１.
⇒ topological charge is conserved !!
We have a geometrical definition of the
topological charge;
Plaquette action(β=2.0) Luscher’s action(β=0.5)
4) .
(ε＝√２,Q=2.)
Note : the effect of ε is only O(a
2. θvacuum simulation of 2D QED
We
also
a method
to calculate
Thus
wedeveloped
can calculate
the reweighting
factor
RQ (=ZQ/Z0) .



Classical solution
⇒ Constant field strength.
Moduli (constant potential which affects Polyakov
loops ) integral of the determinants
⇒ Householder and QL method.
Integral of
⇒ fitting with polynomiyals.
2. θvacuum simulation of 2D QED
Thus, we could evaluate
..
by Luscher’s gauge action and our reweighting
method and the results were consistent with the
continuum theory.
Details are shown in
HF,T.Onogi,Phys.Rev.D68,074503(2003).
2. θvacuum simulation of 2D QED
For example,
we calculated
the pseudoscalar
Pseudo scalar
condensates
in each sector
condensates,
which can be obtained from the anomaly equation,
Our data (using DWF.) are consistent with Q/mV !!
2. θvacuum simulation of 2D QED
We evaluate
total
expectation＜iψγ
value５ψ＞
in θ
Pseudo the
scalar
condensates
θvacuum;
does condense !!
Details are in HF,T.Onogi, Phys.Rev.D70,054508(2004).
The dashed line:Y.Hosotani and R.Rodoriguez,J.Phys.A31,9925(1998)
3. How to fix topology in 4D QCD
Also in 4D QCD, we expect that
obtained with the action;
can be
but…
 No
geometrical (and practical ) definition of Q.
 The
index of D ⇒ a lot of computational costs.
3. How to fix topology in 4D QCD

(New) Cooling method
We “cool” the configuration smoothly by
performing the hybrid Monre Carlo steps with
decreasing g2 (admissibility is always satisfied.).
⇒ We obtain a “cooled ” configuration
close to the classical background at
very weak g2～0.0001 then,
gives an integer with 10% accuracy.
3. How to fix topology in 4D QCD
 How
be
?
G determined
Letcan
usεSsearch
the parameters;
ε< 1/20.49
β, ε, V, Δτ …
ε= 1.0
which can fix Q.
The stability of Q is proved only when
Anothor
group is .also studying this action.
ε＜ εmax ～1/20.49
Q=0
ε=∞
Q=1 S.Necco
S.Shcheredin, W.Bietenholz,
K.Jansen, K.I.Nagai,
⇒
andconfiguration
L.Scorzato, hep-lat/0409073
space is too narrow
4～204 lattice.
on
10
If the barrier is high enough, Q may be fixed.
3. How to fix topology in 4D QCD

Numerical simulations







..
Action: Luscher action (quenched)
Algorithm : The hybrid Monte Carlo method.
Gauge coupling : β= (6/g2) = 1.0 ～ 2.8
Lattice size : 104,144,164
Admissibility condition : 1/ε= 2/3 ～ 1.0
Topological charge
: Q=-３～+３
10 ～ 40 molecular dynamics steps with the step
size Δτ= 0.001-0.02 in one trajectory.
The simulations were done on the
Alpha work station at YITP and SX-5
at RCNP.
3. How to fix topology in 4D QCD

Initial configuration
For topologically non-trivial initial configuration,
we use a discretized version of instanton
solution on 4D torus; A.Gonzalez-Arroyo,hep-th/9807108,
M.Hamanaka,H.Kajiura,Phys.Lett.B551,360(‘03)
which gives constant field strength with
topological charge Q.
3. How to fix topology in 4D QCD

Results
1/ε=1.0
1/ε=2/3
Topology fixing for 2000 trajectory length
β= 1.0
V = 104 Δτ=0.001
×
β= 1.25
V = 10
Δτ=0.001
×
β= 1.4
V = 104 Δτ=0.004
○
β= 1.5
V = 104 Δτ=0.004
○
β= 1.55
V = 104 Δτ=0.004
○
β= 1.5
V = 144 Δτ=0.004
○
β= 2.4
V = 104 Δτ=0.003
×
β= 2.5
V = 104 Δτ=0.003
×
β= 2.6
V = 104 Δτ=0.008
×
β= 2.6
V = 104 Δτ=0.003
○
β= 2.7
V = 104 Δτ=0.003
○
β=2.4,
β=2.6,
β=1.25,
β=1.5,
1/ε=2.0/3.0
1/ε=1.0
1/ε=1.0
4
1/ε=0.0
β= 6.0 length
V ==100
104 Δτ=0.001
1000
trajectory
uncorrelated ×
confs
3. How to fix topology in 4D QCD

Physical scale of the lattice
Wilson loops
Both 1/ε=1.0, β≧1.4 and 1/ε=2/3, β≧2.6
are corresponding β≧6.0 （1/a ～2.0 GeV）
for plaquette action.
⇒ Q can be fixed on lattices finer
than (2.0GeV)-1.
Data of plaquette action are from H.Matsufuru’s web page.
4. Tempering for QCD θvacuum
Our method to calculate the partition
function used in 2D QED;
is not applicable to 4D QCD due to its
large numerical costs.
4. Tempering for QCD θvacuum

Extended
configuration space
Tempering
method
Let us extend
configuration
space;
Luscherthe
action
plaquette
action
Q=0
↓
Q=1
If we obtain N0,
Q = -1
：
↓
N1,
↓ Q is not
conserved.
N-1, ・・・ samples,
4. Tempering for QCD θvacuum

There are several tempering methods...


Simulated tempering G.Parisi et al,Europhys.Lett.19,451(‘92)
Parallel tempering
E.Marianari et al,cond-mat/9612010,
K.Hukushima et al, cond-mat/9512035

…
If the tempering is applicable for Luscher
action, we may able to treat θvacuum,
5. Conclusion

We find at V=104 , 144 ,



Well- controlled topological charge can be measured
by our cooling method .
Q can be fixed for more than 2000 trajectory
length (200 uncorrelated samples .) if the lattice is
fine enough (1/a ～ 2.0GeV) even when we set
1/ε=2/3.
Next we will try





more quantitative studies
large volume
The index of GW fermion
tempering for θvacuum
full QCD
6. Outlook

Luscher’s ‘admissibility’ condition will be
important for …
 Locality
of Dirac operators
 ε-regime
 Finite temperatures and Chiral transition
 SUSY
 Majorana fermions?
 Non commutative lattice ??
 Matrix model ???
…